Multivariate Analysis I

HES 505 Fall 2022: Session 19

Matt Williamson

Objectives

By the end of today you should be able to:

  • Define the weights of evidence approach to overlays

  • Recognize the link between regression analysis and overlay analysis

  • Generate spatial predictions based on regression analysis

  • Extend logistic regression to presence-only data models

Estimating favorability

\[ \begin{equation} F(\mathbf{s}) = \prod_{M=1}^{m}X_m(\mathbf{s}) \end{equation} \]

  • Treat \(F(\mathbf{s})\) as binary
  • Then \(F(\mathbf{s}) = 1\) if all inputs (\(X_m(\mathbf{s})\)) are suitable
  • Then \(F(\mathbf{s}) = 0\) if not

Estimating favorability

\[ \begin{equation} F(\mathbf{s}) = f(w_1X_1(\mathbf{s}), w_2X_2(\mathbf{s}), w_3X_3(\mathbf{s}), ..., w_mX_m(\mathbf{s})) \end{equation} \]

  • \(F(\mathbf{s})\) does not have to be binary (could be ordinal or continuous)

  • \(X_m(\mathbf{s})\) could also be extended beyond simply ‘suitable/not suitable’

  • Adding weights allows incorporation of relative importance

  • Other functions for combining inputs (\(X_m(\mathbf{s})\))

Weighted Linear Combinations

\[ \begin{equation} F(\mathbf{s}) = \frac{\sum_{i=1}^{m}w_iX_i(\mathbf{s})}{\sum_{i=1}^{m}w_i} \end{equation} \]

  • \(F(s)\) is now an index based of the values of \(X_m(\mathbf{s})\)

  • \(w_i\) can incorporate weights of evidence, uncertainty, or different participant preferences

  • Dividing by \(\sum_{i=1}^{m}w_i\) normalizes by the sum of weights

Model-driven overlay

\[ \begin{equation} F(\mathbf{s}) = w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon \end{equation} \]

  • If we estimate \(w_i\) using data, we specify \(F(s)\) as the outcome of regression

  • When \(F(s)\) is binary → logistic regression

  • When \(F(s)\) is continuous → linear (gamma) regression

  • When \(F(s)\) is discrete → Poisson regression

  • Assumptions about \(\epsilon\) matter!!